1. Field of the Invention
The present invention relates to a data conversion method based on negative β-map, and particularly to an A/D converter and a chaos generator using a discrete time integrator for adapting to an integrated circuit.
2. Description of the Related Art
Conventionally, the data conversion method based on negative β-map using a negative real number as the radix has been proposed (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below). In this method, the conversion errors near the ends of tolerance of a threshold are mitigated particularly, over the conventional data conversion methods based on β-map using a positive real radix (see Patent Document 2, and Non-Patent Documents 3 and 4 listed below). This is because while the β-map using a positive real number as the radix provides for the constant size of an invariant subinterval that translates within a domain depending on the value of a threshold parameter, the negative β-map provides for an invariant subinterval positioned substantially at the center of a domain and its size expands or contracts depending on the value of a threshold parameter, allowing a wider dynamic range of a circuit to be obtained for the expanded domain.
Now, detailed description will be provided.
The negative β-map R (•) used in a data converter based on negative β-map is described in Equation (1) (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below).
                              R          ⁡                      (            x            )                          =                  {                                                                                          s                    -                                          β                      ⁢                                                                                          ⁢                      x                                                        ,                                                                              x                  ∈                                      [                                          0                      ,                                              γ                        ⁢                                                                                                  ⁢                        v                                                              )                                                                                                                                                                  β                      ⁢                                                                                          ⁢                      s                                        -                                          β                      ⁢                                                                                          ⁢                      x                                                        ,                                                                              x                  ∈                                      [                                                                  γ                        ⁢                                                                                                  ⁢                        v                                            ,                      s                                        )                                                                                                          (        1        )            wherein, νε[s(β−1), s) is a threshold parameter, −2<−β<−1 is a conversion radix, and γ=1/β, s>0 is a scaling constant.
Assuming that the discrete time is tn (n is a natural number), and then by using it to rewrite Equation (1) above as the one-dimensional discrete time dynamical system, it can be described as
                              x          ⁡                      (                          t                              n                +                1                                      )                          =                              R            ⁡                          (                              x                ⁡                                  (                                      t                    n                                    )                                            )                                =                      {                                                                                                      s                      -                                              β                        ⁢                                                                                                  ⁢                                                  x                          ⁡                                                      (                                                          t                              n                                                        )                                                                                                                ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              0                        ,                                                  γ                          ⁢                                                                                                          ⁢                          v                                                                    )                                                                                                                                                                                      β                        ⁢                                                                                                  ⁢                        s                                            -                                              β                        ⁢                                                                                                  ⁢                                                  x                          ⁡                                                      (                                                          t                              n                                                        )                                                                                                                ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                                                        γ                          ⁢                                                                                                          ⁢                          v                                                ,                        s                                            )                                                                                                                              (        2        )            An example of the one-dimensional map is shown in FIG. 16. In this figure, β=5/3, s=3, ν=5/2, and γν=3/2. FIG. 16 also illustrates the trajectory with x(t1)=0.6 as an initial value. As shown in FIG. 16, the trajectory is resultantly confined within an invariant subinterval [LB, UB) (the region D in FIG. 16). Here, LB and UB are given in Table 1.
TABLE 1Range of νLBUB            (              β        -        1            )        ⁢    s    ≦  ν  <                              β          2                -        β        +        1                    β        +        1              ⁢    s  βν − (β2 − β)sβs − ν                               β          2                -        β        +        1                    β        +        1              ⁢    s    ≦  ν  <                              2          ⁢          β                -        1                    β        +        1              ⁢    s  s − νβs − ν                               2          ⁢          β                -        1                    β        +        1              ⁢    s    ≦  ν  <  ss − νβν − (β2 − β)s
From this table, it is found that the size of the invariant subinterval is maximized when ν=(β−1)s and ν=s, where LB=0 and UB=s. That is, the size of the invariant subinterval is maximized when ν takes the maximum or minimum value within the tolerance of ν, [s (β−1), s). When implementing this map in a circuit, the larger the invariant subinterval is, the wider the dynamic range of the circuit can be, improving the S/N ratio relatively. It should be noted however that the conversion errors vary depending on the value of ν (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below). Moreover, the size of invariant subinterval requires to be set up so that the trajectory of the map does not run off the domain [0,s) due to noises or non-ideal characteristics of the circuit.
Next, a binary variable b(tn)ε{0,1} is defined as
                              b          ⁡                      (                          t              n                        )                          =                                            Q              θ                        ⁡                          (                              x                ⁡                                  (                                      t                    n                                    )                                            )                                =                      {                                                                                0                    ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              0                        ,                        θ                                            )                                                                                                                                        1                    ,                                                                                                              x                      ⁡                                              (                                                  t                          n                                                )                                                              ∈                                          [                                              θ                        ,                        s                                            )                                                                                                                              (        3        )            wherein, Qθ(•) is a quantizer with θ as the threshold. Also, the following applies hereinbelow.θ=γν  (4)At this time, Equation (2) above can be described asx(tn+1)= b(tn)s+b(tn)βs−βx(tn)=s( b(tn)+βb(tn))−βx(tn)  (5)This equation can be further transformed as
                                                                        x                ⁡                                  (                                      t                                          n                      +                      1                                                        )                                            =                            ⁢                                                s                  ⁡                                      (                                                                                            b                          ⁡                                                      (                                                          t                              n                                                        )                                                                          _                                            +                                              β                        ⁢                                                                                                  ⁢                                                  b                          ⁡                                                      (                                                          t                              n                                                        )                                                                                                                )                                                  -                                  β                  ⁢                                                                          ⁢                                      x                    ⁡                                          (                                              t                        n                                            )                                                                                                                                              =                            ⁢                                                s                  ⁡                                      (                                                                                            b                          ⁡                                                      (                                                          t                              n                                                        )                                                                          _                                            +                                                                        (                                                      1                            +                            β                            -                            1                                                    )                                                ⁢                                                  b                          ⁡                                                      (                                                          t                              n                                                        )                                                                                                                )                                                  -                                  β                  ⁢                                                                          ⁢                                      x                    ⁡                                          (                                              t                        n                                            )                                                                                                                                              =                            ⁢                                                s                  ⁡                                      (                                                                                            b                          ⁡                                                      (                                                          t                              n                                                        )                                                                          _                                            +                                              b                        ⁡                                                  (                                                      t                            n                                                    )                                                                    +                                                                        (                                                      β                            -                            1                                                    )                                                ⁢                                                  b                          ⁡                                                      (                                                          t                              n                                                        )                                                                                                                )                                                  -                                  β                  ⁢                                                                          ⁢                                      x                    ⁡                                          (                                              t                        n                                            )                                                                                                                              (        6        )                                                          ⁢                  =                                    s              ⁢                              (                                                      b                    ⁡                                          (                                              t                        n                                            )                                                        +                                      b                    ⁡                                          (                                              t                        n                                            )                                                        +                                                            (                                              β                        -                        1                                            )                                        ⁢                                          b                      ⁡                                              (                                                  t                          n                                                )                                                                                            )                                      -                          β              ⁢                                                          ⁢                              x                ⁡                                  (                                      t                    n                                    )                                                                                        (        7        )            
Here, assume that the input signal xinput is sampled at t=t1. That is,x(t1)=xinput  (8)At this time, by repeating Equation (5) above from t=t1 to t=tL is a bit length after A/D conversion), the binary signal train BS (xinput) corresponding to the input signal xinput is obtained.BS(xinput)=(b1b2 . . . bL)−β,s  (9)wherein, bn=b(tn) (n=1, 2, . . . , L), bL=b(tL) is the LSB (least significant bit), and b1=b(t1) is the MSB (most significant bit).
At this time, in order to obtain a decoded value {circumflex over (x)}L of an original signal from the output bit series of L-bits,
                                          x            ^                    L                =                  s          ⁢                      {                                                                                (                                          -                      γ                                        )                                    L                                2                            -                                                ∑                                      i                    =                    1                                    L                                ⁢                                                      (                                                                                            b                          i                                                ⁢                        β                                            +                                                                        b                          i                                                _                                                              )                                    ⁢                                                            (                                              -                        γ                                            )                                        i                                                                        }                                              (        10        )            may be applied (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below).
Here, the tolerance σv of the threshold parameter ν of the quantizer Qθ(•) is given as follows with s and β (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below).σν=s(2−β)  (11)This is shown in FIG. 16 as the bold line on the axis x(tn+1). Accordingly, the threshold θ of the quantizer is allowed to vary within the range described as follows (see Patent Document 1, and Non-Patent Documents 1 and 2 listed below).σθ=γσν=γs(2−β)=s(2γ−1)  (12)This is shown in FIG. 16 as the bold line on the axis x(tn). That is, even when the quantization threshold θ varies due to the change in environment, non-ideal characteristics of circuit elements, or noises, the A/D converter circuit operates normally as long as θ remains within the range described in Equation (12) above. In other words, the A/D converter circuits that operate normally can be realized, even with the simple and inexpensive circuit configuration where the quantization threshold θ may vary.
The configuration diagrams of the A/D converter based on negative β-map are shown in Patent Document 1, and Non-Patent Documents 1 and 2 listed below. FIG. 17 shows a configuration diagram of the A/D converter based on negative β-map. However, the as-is configuration is not suitable for integrated circuits.    Patent Document 1: WO 2010/024196    Patent Document 2: WO 2009/014057    Non-Patent Document 1: S. Hironaka, T. Kohda, and K. Aihara, “Negative β-encoder”, in Proceedings of International Symposium on Nonlinear Theory and Its Applications, pp. 564-567, Budapest, Hungary, 2008    Non-Patent Document 2: T. Kohda, S. Hironaka, and K. Aihara, “Negative β-encoder”, Preprint, arXiv:0808.2548v2[cs.IT], 28 Jul. 2009, http://arxiv.org/abs/0808.2548    Non-Patent Document 3: T. Daubechies, R. A. DeVore, C. S. Gunturk, and V. A. Vaishampayan, “A/D conversion with imperfect quantizers”, IEEE Transactions on Information Theory, Vol. 52, No. 3, pp. 874-885, 2006    Non-Patent Document 4: S. Hironaka, T. Kohda, and K. Aihara, “Markov chain of binary sequences generated by A/D conversion using β-encoder”, in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic Systems, pp. 261-264, Tokushima, Japan, 2007